We study maximalblocks of consecutive primeterms in A007310, the sequence of positive integers congruent to 1 or 5 modulo 6. We prove that the initial block 5, 7, 11, 13, 17, 19, 23 is the unique prime block of length 7, and that no longer prime block exists. For each r≥1, we define Pr as the sequence of r-th terms of the prime blocks. As consequences, P7 = 23 and Pr = ∅ for all r≥8.
Stephen Steiner (Fri,) studied this question.